Looking for Gröbner basis theory for (almost) skew 2-nomial algebras
Looking for Gröbner basis theory for (almost) skew 2-nomial algebras
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DOI：
10.1016/j.jsc.2010.05.002
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发表时间：
2008-08
期刊：
影响因子：
--
通讯作者：
Huishi Li
中科院分区：
文献类型：
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作者：
Huishi Li
在本文中，我们引入（几乎）偏斜 2-nomial 代数，并在适度的水平上为此类代数寻找单侧或两侧的 Gröbner 基础理论。也就是说，我们为每个偏斜 2 项式代数建立了偏斜乘法 K 基的存在性，并且我们探索了（几乎）偏斜 2 项式代数的（左、右或两侧）单项式排序的存在性。与普遍认可的具有 Gröbner 基理论的代数（例如可解类型的代数（Kandri-Rody 和 Weispfenning，1990）及其一些同态图像）不同，偏斜 2-项式代数的子类具有左 Gröbner 基理论，但不一定具有两侧 Gröbner 基理论，分别是偏斜的子类 具有正确的 Gröbner 基础理论但不一定具有双面 Gröbner 基础理论的 2 项式代数被确定为使得许多量子二项式代数（为 Yang-Baxter 方程提供二项式解）及其 Koszul 对偶（Gateva-Ivanova 和 Van den Bergh，1998；Laffaille，2000；Gateva-Ivanova，2009）为 涉及。
In this paper, we introduce (almost) skew 2-nomial algebras and look for a one-sided or two-sided Gröbner basis theory for such algebras at a modest level. That is, we establish the existence of a skew multiplicative K-basis for every skew 2-nomial algebra, and we explore the existence of a (left, right, or two-sided) monomial ordering for an (almost) skew 2-nomial algebra. As distinct from commonly recognized algebras holding a Gröbner basis theory (such as algebras of the solvable type (Kandri-Rody and Weispfenning, 1990) and some of their homomorphic images), a subclass of skew 2-nomial algebras that have a left Gröbner basis theory but may not necessarily have a two-sided Gröbner basis theory, respectively a subclass of skew 2-nomial algebras that have a right Gröbner basis theory but may not necessarily have a two-sided Gröbner basis theory, are determined such that numerous quantum binomial algebras (which provide binomial solutions to the Yang–Baxter equation) and their Koszul dual (Gateva-Ivanova and Van den Bergh, 1998; Laffaille, 2000; Gateva-Ivanova, 2009) are involved.